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Abstract

Theoretical demonstration and experimental evidence of photon Bloch oscillations and
Wannier-Stark ladders (WSLs) in dual-periodical (DP) multilayers, based on porous
silicon, are presented. An introduction of the linear gradient in refractive indices
in DP structure, which is composed by stacking two different periodic substructures
N times, resulted in the appearance of WSLs. Theoretical time-resolved reflection spectrum
shows the photon Bloch oscillations with a period of 130 fs. Depending on the values
of the structural parameters, one can observe the WSLs in the near infrared or visible
region which may allow the generation of terahertz radiation with a potential applications
in several fields like imaging.

Keywords:

Background

The analogs between electron transport and propagation of the optical waves in dielectric
structures opened the possibility of the implementation of Bloch oscillations for
electromagnetic waves in photonic crystals (PC) [1,2]. The photonic analog of the above-mentioned effect appears when a PC is subjected
to a slowly varying refractive index or a geometric parameter modulation, resulting
in a linear tilting of the band structure. Such ‘chirped’ PCs give rise to a set of
equidistant frequency levels [3], i.e., the optical counterpart of the Wannier-Stark ladders (WSLs) in semiconductor
superlattices. Recently, different methods have been adopted to tilt the photonic
band for the observation of WSLs and photon Bloch oscillations (PBOs) [4-7]. In confined Bragg mirrors, the band structure modifications are due to the gradual
change of the lateral confinement [4]. On the other hand, in geometrically chirped PCs, the band structure modulation arises
from a gradual increase in the thicknesses of the layers [5,6]. Furthermore, in graded-index optical superlattices, the index gradient comes from
a linear modification in the refractive indices of the layers [7,8], and so on. The use of above-mentioned PCs with gradient in optical thickness has
evoked special interests in many applications due to their novel properties [9]. Such multilayer structures originate a new type of Fabry-Perot cavity where the
reflectors are replaced by nonpropagating regions associated with the local periodicity
of the structure. If the linear gradient is considered, the distance between the band
edges where the PBOs occur can be maintained constant. Hence, the period of PBOs remains
constant with the change in the frequency of the incident wave. In case of nonlinear
gradient, the distance between the band edges and, therefore, the period of the resulting
PBOs can be tuned by changing the frequency of the incident light. For example in
2005 Lousse and Fan [8] reported the tunable terahertz Bloch oscillations in the chirped photonic crystals,
with the potential applications in several fields, like biomedical sensing. Such useful
photonic structures can be fabricated with different materials. Recently, one-dimensional
photonic superlattices made of porous silicon (PSi) have allowed the demonstration
of optical analogs [10] of electronic phenomena [11,12], such as PBOs, Zener tunneling, and Anderson localization [6,7,13,14].

In the present work, we demonstrate the theoretical and experimental evidence of WSLs,
using dual-periodical multilayer structures (shown schematically in Figure 1), with a linear gradient in refractive indices, based on porous silicon. Theoretical
evidence of the presence of PBOs in such structures is also presented.

Figure 1.Dual-periodical multilayer structure. Schematic view of the 1D dual-periodic structure showing the layer parameters, where
n and m represent the period numbers of a and b in A and B substructures, respectively; n1 and n2 are high and low refractive indexes of alternating dielectric layers in a and b; the layer thickness is d1 and d2 for a, and d3 and d4 for b, respectively.

Dual-periodic structures

The optical properties of dual-periodical (DP) structures have been theoretically
and experimentally reported by several groups [32,33]. Recently, Pérez et al. [34] reported DP structures from PSi multilayers. Dual-periodic structure (Figure 1) is composed of two substructures, A and B, repeating alternatively in the sequence . The An and Bmare in turn composed of two different periodic units, a and b, respectively, where subscripts, n and m, are the number of periods for a and b in the A and B substructures, respectively. Both a and b consist of a pair of layers with high and low refractive indices. The thickness of
the double layer a is da = d1 + d2d1 and d2 being the thicknesses for the layers with the high (n1) and low (n2) refractive indices, respectively. Similarly, the double layer b has thickness db = d3 + d4; d3 and d4 being the thicknesses for layers with the high and low refractive index as well.
In particular, the following sequence was used: A2B4A2B4A2B4A2B4A2B4A2B4= for the infrared region. If the substructure B is considered as a defective layer, the frequency intervals where the resonances
of the transmission peaks appear can be reduced by increasing the number of periods
a in the substructure A. On the other hand, if substructure A is a defective layer, the frequency intervals of the resonances can be increased
by reducing the number of periods b in the substructure B. When identical AnBmstructures are coupled, a degenerate mode repulsion arises. Each degenerate optical
resonance splits up and a miniphotonic band forms [32-34]. Due to the periodicity of the structure, the miniphotonic bands are separated by
photonic band gaps in which propagation is prohibited. Moreover, when N>1, there are N−1 defect layers; therefore, N−1 resonance modes and N−1 transmission peaks will appear in the spectra. By adjusting the structural parameters,
it is possible to tune the number, frequency, and full width at half-maximum (FWHM)
of the resonance modes, opening the possibility to fabricate optical filters based
on porous silicon multilayers. Such DP photonic structures are very promising in the
field of optoelectronics, optical communications, and optical biosensors [34].

Furthermore, 1D translational symmetry of the system should be broken by introducing
a small gradient in the refractive indices along the depth of the DP structure to
obtain PBOs in periodic 1D photonic crystals. The gradient in the refractive indices
results in a spatial tilting of the miniband and photonic band gaps in which the resonances,
due to defects in DP structure, change slightly while preserving the mode coupling.
In this way, the extended photonic states are turned into a discrete sequence of energy
levels with level spacing △E, which is an optical equivalent of a WSL in frequency
domain. The refractive index gradient in layers is given by △n = ( - )/, where the subscripts z1and zm are the first and the m-th layer along the depth within the sample. This gradient
is the optical counterpart of the external electric field used in electronic superlattices.

Methods

The structures were fabricated by electrochemical etching of boron-doped silicon wafers
with resistivity from 0.007 to 0.013 Ω· cm and (100) orientation. The substrates were etched in an electrolyte consisting
of HF (40%) and ethanol (99.98%) in the volumetric ratio of 1:1. To estimate the refractive
index corresponding to a given current density, the effective medium Bruggeman’s model
is used. To measure the porosity of the layers, gravimetrical method was employed.
The reflectivity spectra were measured using p-polarized transverse-magnetic (TM)
modes light at an incidence angle of 20° using VARIAN-CARY 5000 spectrometer (Varian
Inc., NC, USA ). Two different sets of structural parameters were used to fabricate
the PSi structures to observe WSLs and PBOs in the near infrared region. The experimental
reflectivity results were compared with the theoretical simulations. To obtain structure with resonance transmission modes in the near infrared region d1 = 105, d2 = 315, d3 = 216, and d4 = 103 nm were considered. Their refractive indices were taken as n1= 2.2 and n2 = 1.4, which correspond to the porosities of 48% and 76%, and are obtained with current
densities (J) of 29 and 134 mA/cm2, respectively. For obtaining a linear refractive index gradient of 16% in structure (named as G(A2B4)6), n1 changes from 2.2 to 2.55 (37% porosity, J = 5.2 mA/cm2) and n2 from 1.4 to 1.57 (69% porosity, J = 89.0 mA/cm2).

Theoretical model

Light propagation in dual-periodical multilayers

Photon propagation in one-dimensional structures has been modeled by simple transfer
matrix method considering p-polarized light [35]. If we consider an electromagnetic (EM) wave propagating in the structure with propagation
constant k=k∥ + kzẑ, there are two independent EM modes: TM and transverse electric (TE). The electric
and magnetic field for the TE and TM mode are perpendicular to the plane defined by
the wave vector and the direction of periodicity. Using the transfer matrix method,
we can relate the amplitudes of the fields and in the j-th cell of the system to the amplitudes of the field in the (j + 1)-th cell
according to [36],

(1)

where and is the amplitude of the wave in layer j with polarization μ= sptraveling to the right and to the left, respectively. For the case considered in this
work, the total transfer matrix of the system can be written as a product of matrices
of the type [36]:

(2)

where qjμ = for p-polarization and qjμ = kjz for s-polarization; ϕj=kjzdjkj=(ω/c)njkjz being the component of the wave vector along the growth direction of the system in
the j-th layer given by ; and , its complex refractive index. A real approach over the transfer matrix was implemented
by considering the absorption and dispersion dependence. Reflectivity calculations
of the system are given in terms of the matrix elements of the total transfer matrix
according to . Using transfer matrix calculations, it is possible to compute the electric field
distribution inside the structure. The electric field in the structure along the z direction for a certain wavelength can be expressed as follows:

(3)

where are the elements of the transmission matrix from the first to the k-th interface.
As can be seen from this expression, the reflectivity coefficient of the whole structure
must be calculated previously before evaluating the field amplitude inside the structure.
Hence, the intensity is simply the square of the electric field, |E|2 which is normalized to the amplitude of the incident signal. The amplitude of the
time-resolved reflection can be calculated using the following expression:

(4)

where r(ω) is the reflectivity coefficient, and g(ω) is the incident pulse with a Gaussian spectral function in frequency domain:

(5)

where E0 is the central photon energy of the pulse and δ is the pulse width. The parameter δcontrols the pulse duration, which should be sized to a value close to the Bloch oscillations
periods. The expression in Equation 4 is valid only for times longer than the duration
of the incident pulse [6].

Porous silicon as an effective medium

As PSi layer consists of two distinct components (air and silicon); its complex dielectric
function (εPSi) has an intermediate value between silicon (εSi) and air (εair), i.e., a volume fraction fp of silicon and a volume fraction (1 - fp) of pores (where refractive index of air = 1). Bruggeman effective medium approximation
was used to estimate the effective optical parameters of PSi [37,38]:

(6)

From this equation we obtain εPSiin terms of εSi and the porosity fP:

(7)

(8)

The refractive index n(λ) and the extinction coefficient k(λ) can be obtained by the Cauchy model, which is useful for dielectric materials (with
exponential absorption), far from the absorption bands [39].

Results and discussion

Figure 2a shows the square of the electric field inside the DP structure, without any refractive index gradient (△n = 0), calculated by transfer matrix methods using Equation 3 for infrared region.
The scattering states map provides the electric field intensity at each position inside
the structure when plane electromagnetic waves of unit amplitude impinge onto the
structure to 20° of incidence. A change from dark to bright corresponds to the increase
in the light intensity. Although, the input intensity has been normalized to unity,
in some of the regions, the intensity inside the structures can be more than one due
to internal resonances. No field localization is observed among the different zones
in the DP structure. Composed by some delocalized states in space, each bright band
is an optical analog of electronic energy band caused by the periodic potential in
electronic superlattices. In Figure 2b the optical reflectivity measurements performed on DP structure, fabricated for
infrared region of the electromagnetic spectrum, are presented. The dotted lines in
Figure 2b correspond to the results obtained by numerical simulations. The measured reflectance
spectrum shows the presence of five resonance peaks at photons energy 0.837, 0.853,
0.870, 0.885, and 0.896 eV, with FWHM of 6, 8.2, 8, 7.1, and 4 meV, respectively.
The FWHMs of the resonance peaks can be tuned by increasing or decreasing the number
of layers in either of the substructures.

Figure 2.Scattering states map and reflectance spectra of (A2B4)6structure. (a) Calculated electric field intensity distribution of TM modes in (A2B4)6 structure with no refractive index gradient i.e., △n= 0 (flat miniband situation). A change of color from dark to bright corresponds to
the increase in the light intensity. (b) Measured and simulated reflectance spectra of the structure. The schematic illustration
of the corresponding DP structure is shown just above the scattering states map.

The decreasing or the increasing number of layers of the substructure, acting as a
defect in the system (substructure B for the samples considered here), results in the widening or narrowing of the peaks
and consequently, the value of FWHM [34].

Figure 3a shows the square of the electric field inside the G structure with △n = 16% in the same range of frequencies as in Figure 2. One can clearly observe the light confinement due to the inclination of the minibands
and photonic bandgaps (PBGs) (Figure 3a). The occurrence of WSL as a series of narrow resonant peaks is clearly observed
in the reflectivity spectra of the structure (Figure 3b). When the incident pulse enters in G structure in the frequency region between two local gaps, most of the light is reflected
back on the lower band gap, and only a small percentage which crosses the gap elastically
(Zener tunneling) feeds the horizontal resonances associated with the WSLs. Hence,
the confined light starts to oscillate within the inclined allowed miniband with a
regular period of τB = h/△E. These are PBOs which represent the time domain counterpart of the photonic WSLs.
Time domain oscillations can be detected by measuring the time-resolved reflection
of the sample. The center of the Wannier-Stark resonance defines, in space, the center
of the PBO. The distance between successive frequency levels can be obtained by the
following Fabry-Perot formula: △E = hc/, where d is the local distance between the gap edges at the given input frequency and is the average refractive index. The measured values of △E, from Figure 3b, are found
to be in agreement with the theoretical simulations presented in Figure 3a (as a function of film depth). The structural parameters are modulated to obtain
the energy spacing △E ≈ 27 meV, and therefore, τB = 148 fs.

Figure 3.Scattering states map and reflectance spectra of G(A2B4)6structure. (a) Calculated electric field intensity distribution of TM modes in G(A2B4)6 structure with a linear gradient in refractive index (△n = 16%). Inclined white lines are shown as indicators of the miniphotonic band inclination.
One can observe the formation of the photonic WSLs confined between two minigaps.
(b) Measured and simulated reflectance spectra of the structure and the input Gaussian
pulse (red dotted line) in frequency domain.

Figure 4 shows the theoretical time-resolved reflection of the G structure calculated using Equation 4 for a Gaussian pulse with E0= 809 meV (δ = 10.5 meV), as shown in Figure 3b, and E0= 830 meV (δ= 11 meV). One can observe the PBOs with an oscillation period of 111 and 130 fs(Figure
4), compared to 148 fs (△E = 27 meV) measured from the reflectance spectrum of the
corresponding structure (Figure 3b). These oscillations are observable due to the tunneling of photons through the
lower inclined gap. The form and the energy of the pulse are found to slightly influence
the period of the PBOs, e.g., an increase in the pulse energy by 21 meV results in
a increase of PBO period by 19 fs.

Conclusions

Successful demonstration of WSLs and consequently the PBOs in one-dimensional dual-periodical
PSi structures for near infrared range is presented. The possibility of observing
the WSLs in reflectance spectrum is evaluated by selecting the adequate parameters
with a linear gradient in refractive indices. Such photonic structures can be very
promising in the observation of Bloch oscillations in the different regions of the
electromagnetic spectrum.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JOE prepared the PSi multilayers. JA, ER-A, and JE performed the treatment of experimental
data and simulations. AMB and VA proposed the experimental conditions and measured
the structures. JOE, JA,and VA prepared the manuscript initially and participated
in its design and coordination. All authors read and approved the final manuscript.

Acknowledgements

This work has been partially supported by CONACyT under project 128953 (for VA) and
101486 (for AMB).